CAREER: Locally Homogeneous Geometric Manifolds and Their Moduli Spaces

这个项目涉及局部齐次几何流形。 这些是抽象的数学对象，旨在模拟我们生活的宇宙。 术语局部齐性是指存在高度的局部对称性，其被称为（局部）对称群的李群捕获。 正是这个对称群支配着几何，在以下意义上：我们在几何流形中可以测量的有意义的量（例如长度或角度）正是那些在对称群下不变的量。 有许多不同的可能的对称群，导致不同类型的几何流形在数学和物理的许多背景下都很有用。 也可以有许多不同的几何流形具有相同的局部对称群。这些都具有相同的局部属性，但在大尺度上看起来可能非常不同。所有这些可能性的空间被称为模空间;它是一个拓扑空间，其点是某种类型的可能几何流形，其拓扑将这些几何流形组织成特征连续变化的族。虽然精确的特征（例如形状、大小等）我们的宇宙是一个经验物理学的问题，模空间是宇宙可能具有什么特征的数学答案。 该项目的研究部分研究许多类型的几何流形，重点是平坦仿射几何，真实的射影几何和常曲率伪黎曼几何。这些几何有着恰到好处的对称性，使得大而有趣的模空间具有神秘但易处理的行为。教育部分的核心要素包括培养博士。一个新的德州实验几何实验室的本科生研究，和德州冬季讲习班几何结构的早期职业数学家。在PI的研究计划的指导思想是，一种类型的几何流形可以卓有成效地研究变形，或过渡到不同类型的几何。 例如，PI的工作研究马古利斯时空作为反德西特（AdS）时空的几何极限，已经产生了许多关于这些仿射洛伦兹三维流形的几何，拓扑和变形理论的结果。PI将开发新的工具，以下几何极限的观点，为新的仿射几何背景下，在更高的维度，以解决问题，围绕重要的开放问题，如Auslander猜想。PI还将开展一项广泛的计划，研究流形上的凸真实的投影结构。其中一个重点是确定哪些三流形承认这样的结构，并描述模空间，与低维拓扑问题的应用。在更一般的情况下，PI将进行一个新的概念，凸余紧性在真实的射影几何的彻底调查，推广了良好的研究概念从克莱因集团设置。该项目的教育目标是扩大和垂直整合UT奥斯汀几何结构研究培训计划，由PI开发。德州实验几何实验室（TXGL）将介绍本科生在UT奥斯汀通过计算和实验项目在几何，拓扑和/或动力学的研究。TXGL还将为博士提供机会。学生和博士后获得经验的指导和传统的课堂环境之外的本科生教学。此外，TXGL将产生可视化，说明当前数学研究的概念。德克萨斯州冬季几何结构研讨会将汇集早期职业数学家的小组，以学习不同（但相关）数学领域交叉的新兴新课题。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project concerns locally homogeneous geometric manifolds. These are abstract mathematical objects designed to model the universe we live in. The term locally homogeneous refers to the presence of a high degree of local symmetry which is captured by a Lie group called the (local) symmetry group. It is this symmetry group which governs the geometry, in the following sense: the meaningful quantities we can measure in a geometric manifold (e.g. lengths or angles) are exactly those which are invariant under the symmetry group. There are many different possible symmetry groups which lead to different types of geometric manifolds useful in many contexts across mathematics and physics. There can also be many different geometric manifolds with the same local symmetry group. These all have the same local properties, but can look very different at large scales. The space of all such possibilities is called a moduli space; it is a topological space whose points are the possible geometric manifolds of a certain type and whose topology organizes those geometric manifolds into families whose features vary continuously. While the precise features (e.g. shape, size, etc.) of our universe is a question for empirical physics, a moduli space is the mathematical answer to the question of what possible features could the universe have. The research component of this project studies geometric manifolds of many types, with focus on flat affine geometry, real projective geometry, and constant curvature pseudo-Riemannian geometry. These geometries have just the right amount of symmetry to allow for large interesting moduli spaces with mysterious but tractable behavior. Core elements of the educational component include training Ph.D. students, a new Texas Experimental Geometry Lab for undergraduate research, and the Texas Winter Workshop on Geometric Structures for early career mathematicians.A guiding philosophy in the PI's research program is that geometric manifolds of one type may be fruitfully studied by deforming, or transitioning, to a different type of geometry. For example, the PI’s work studying Margulis spacetimes as geometric limits of anti de Sitter (AdS) spacetimes, has yielded many results about the geometry, topology, and deformation theory of these affine Lorentzian three-manifolds. The PI will develop new tools, following the geometric limit point of view, for new affine geometry contexts in higher dimensions, in order to address questions surrounding important open problems such as the Auslander Conjecture. The PI will also pursue a broad program to study convex real projective structures on manifolds. One focus is to identify which three-manifolds admit such structures and to describe the moduli spaces, with applications to questions in low-dimensional topology. In a more general context, the PI will conduct a thorough investigation of a new notion of convex cocompactness in real projective geometry, generalizing the well-studied notion from the Kleinian groups setting. Educational goals of this project center on expanding and vertically integrating the research training program in Geometric Structures at UT Austin being developed by the PI. The Texas Experimental Geometry Lab (TXGL) will introduce undergraduates at UT Austin to research in geometry, topology, and/or dynamics through computational and experimental projects. TXGL will also provide opportunities for Ph.D. students and post-docs to gain experience mentoring and teaching undergraduates outside the traditional classroom setting. In addition, TXGL will produce visualizations that illustrate concepts from current mathematics research. The Texas Winter Workshops in Geometric Structures will bring together small groups of early career mathematicians in order to learn an emerging new topic at the intersection of different (but related) mathematical fields.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.